Nearly tight bounds on the number of Hamiltonian circuits of the hypercube and generalizations

It has been shown that for every perfect matching M of the d-dimensional n-vertex hypercube, d?2, n=d², there exists a second perfect matching M^′ such that the union of M and M^′ forms a Hamiltonian circuit of the d-dimensional hypercube. We prove a generalization of a special case of this result when there are two dimensions that do not get used by M. It is known that the number Md of perfect matchings of the d-dimensional hypercube satisfies and, in particular, (2d/n)^n/2(n/2)!?Md?(d!)^n/(2d). It has also been shown that the number Hd of Hamiltonian circuits of the hypercube satisfies 1?limd→∞(logHd)/(logMd)?2. We finally strengthen this result to a nearly tight bound ((dlog2/(eloglogd))(1−on(1)))?Hd?(d!)^n/(2d)((d−1)!)^n/(2(d−1))/2 proving that limd→∞(logHd)/(logMd)=2. This means that the bound Hd?Md is improved to a nearly tight , so the number of Hamiltonian circuits in the hypercube is nearly quadratic in the number of perfect matchings. The proofs are based on a result for graphs that are the Cartesian product of squares and arbitrary bipartite regular graphs that have a Hamiltonian cycle. We also study a labeling scheme related to matchings.     

Wormholes leading to extra dimensions

In 6D general relativity with a scalar field as a source of gravity, a new type of static wormhole solutions is presented: such wormholes connect our universe with a small 2D extra subspace with a universe where this extra subspace is large, and the whole space-time is effectively 6-dimensional. We consider manifolds with the structure M₀ × M₁ × M₂, where M₀ is 2D Lorentzian space-time while each of M_1,2 can be a 2-sphere or a 2-torus. After selecting possible asymptotic behaviors of the metric functions compatible with the field equations, we give two explicit examples of wormhole solutions with spherical symmetry in our space-time and toroidal extra dimensions. In one example, with a massless scalar field (it is a special case of a well-known more general solution), the extra dimensions have a large constant size at the “far end”; the other example contains a nonzero potential V(φ) which provides a 6D anti-de Sitter asymptotic, where all spatial dimensions are infinite.     

Partial Multinode Broadcast and Partial Exchange Algorithms for d-Dimensional Meshes

In this paper we consider the partial multinode broadcast and the partial exchange communication tasks in d-dimensional meshes. The partial multinode broadcast in an N-processor network is the task in which each of M ≤ N arbitrary nodes broadcasts a packet to all the remaining N − 1 nodes. Correspondingly, in the partial exchange there are M ≤ N nodes that wish to send a separate, personalized packet to each of the other nodes. We propose algorithms for the d-dimensional mesh network that execute the partial multinode broadcast and the partial exchange communication tasks in near-optimal time. No assumption is made concerning the locations of the M source nodes. The communication algorithms proposed are "on line" and distributed. We further look at a dynamic version of the broadcasting problem, where broadcast requests are generated at random times. In particular, we assume that the broadcast requests are generated at each node of the mesh according to a Poisson distribution with rate λ. Based on the partial multinode broadcast algorithm, we propose a dynamic decentralized scheme to execute the broadcasts in this dynamic environment. We find an upper bound on the average delay required to serve each broadcast. We prove that the algorithm is stable for network utilization ρ close to 1, and the average delay is of the order of the diameter for any load in the stability region.     

A new matrix test for randomness

Let ${{\mathcal S}}$ be one of the two multiplicative semigroups: M × M Boolean matrices, or the semigroup of M × M matrices over the field GF(2). Then for any matrix ${A\in {\mathcal S}}$ there exist two unique smallest numbers, namely the index and period k, d, such that A ^k  = A ^k+d . This fact allows us to form a new statistical test for randomness which we call the Semigroup Matrix Test. In this paper, we present details and results of our experiments for this test. We use Boolean matrices for M = 2, . . . , 5, and matrices over GF(2) of the size M = 2, . . . , 6. We also compare the results with the results obtained by the well-known Binary Matrix Rank Test.     

Reconfiguration of Rings and Meshes in Faulty Hypercubes

In this paper we present schemes for reconfiguration of embedded task graphs in hypercubes. Previous results, which use either fault-tolerant embedding or an automorphism approach, can be expensive in terms of either the required number of spare nodes or reconfiguration time. Using the free dimension concept, we combine the above two approaches in our schemes which can tolerate about n faulty nodes under the worst case while keeping task migration time small. With expansion-2 initial embedding, three distributed reconfiguration schemes are presented in this paper. The first scheme, applied to chains and rings, can tolerate any ƒ ≤ n − 2 faulty nodes in an n-dimensional hypercube. The second and third schemes are applied to meshes or tori. For a mesh or torus of size 2^m₁ 1 ··· 1 2^m_d, the second scheme can tolerate any ƒ ≤ m_i − 1 faulty nodes, where m_i is the largest direction of the mesh and n = m₁ + ··· + m_d + 1. By embedding two copies of meshes or tori in cube, the third scheme can tolerate any ƒ ≤ n − 1 faulty nodes with the dilation of embedding after reconfiguration degraded to 2. The third scheme is quite general and can be applied to any task graph.     

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×???×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box?(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×⋅⋅⋅×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a ?1+\frac1clogn?^d-1\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1} approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.     

Digital metrics: A graph-theoretical approach

Consider the following two graphs M and N, both with vertex set Z × Z, where Z is the set of all integers. In M, two vertices are adjacent when their euclidean distance is 1, while in N, adjacency is obtained when the distance is either 1 or √2. By definition, H is a metric subgraph of the graph G if the distance between any two points of H is the same as their distance in G. We determine all the metric subgraphs of M and N. The graph-theoretical distances in M and N are equal respectively to the city block and chessboard matrics used in pattern recognition.     

Spectral clustering with more than K eigenvectors

Spectral clustering techniques are heuristic algorithms aiming to find approximate solutions to difficult graph-cutting problems, usually NP-complete, which are useful to clustering. A fundamental working hypothesis of these techniques is that the optimal partition of K classes can be obtained from the first K eigenvectors of the graph normalized Laplacian matrix L_N if the gap between the K-th and the K+1-th eigenvalue of L_N is sufficiently large. If the gap is small a perturbation may swap the corresponding eigenvectors and the results can be very different from the optimal ones.In this paper we suggest a weaker working hypothesis: the optimal partition of K classes can be obtained from a K-dimensional subspace of the first M>K eigenvectors, where M is a parameter chosen by the user. We show that the validity of this hypothesis can be confirmed by the gap size between the K-th and the M+1-th eigenvalue of L_N. Finally we present and analyse a simple probabilistic algorithm that generalizes current spectral techniques in this extended framework. This algorithm gives results on real world graphs that are close to the state of the art by selecting correct K-dimensional subspaces of the linear span of the first M eigenvectors, robust to small changes of the eigenvalues.     

L3: An FPGA-based multilayer maze routing accelerator

This paper describes a multi-layer maze routing accelerator which uses a two-dimensional array of processing elements (PEs) implemented in an FPGA. Routing for an L-layer N×N grid is performed by an array of N×N PEs that time-multiplex each layer over the array. This accelerates the classic Lee Algorithm from O(L×d²) in software to O(L×d). Each PE can be implemented in 32 look up tables in a Xilinx Virtex-II FPGA, which makes possible routing arrays that are large enough to support detailed routing for VLSI. Cycle measurements show a speedup of 50–75× over a 2.54 GHz Pentium 4 for a 4-layer 8×8 array and 93× for a 4-layer 16×16 array.     

Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes

Independent spanning trees (ISTs) on networks have applications to increase fault-tolerance, bandwidth, and security. Möbius cubes are a class of the important variants of hypercubes. A recursive algorithm to construct n ISTs on n-dimensional Möbius cube M _n was proposed in the literature. However, there exists dependency relationship during the construction of ISTs and the time complexity of the algorithm is as high as O(NlogN), where N=2 ⁿ is the number of vertices in M _n and n≥2. In this paper, we study the parallel construction and a diagnostic application of ISTs on Möbius cubes. First, based on a circular permutation n?1,n?2,…,0 and the definitions of dimension-backbone walk and dimension-backbone tree, we propose an O(N) parallel algorithm, called PMCIST, to construct n ISTs rooted at an arbitrary vertex on M _n . Based on algorithm PMCIST, we further present an O(n) parallel algorithm. Then we provide a parallel diagnostic algorithm with high efficiency to diagnose all the vertices in M _n by at most n+1 steps, provided the number of faulty vertices does not exceed n. Finally, we present simulation experiments of ISTs and an application of ISTs in diagnosis on 0-M ₄.     

A new discrete circuit for readout of resistive sensor arrays

In this paper, we present a method that simplifies the interconnect complexity of N × M resistive sensor arrays from N × M to N + M. In this method, we propose to use two sets of interconnection lines in row–column fashion with all the sensor elements having one of their ends connected to a row line and other end to a column line. This interconnection overloading results in crosstalk among all the elements. This crosstalk causes the spreading of information over the whole array. The proposed circuit in this method takes care of this effect by minimizing the crosstalk. The circuit makes use of the concept of virtual same potential at the inputs of an operational amplifier in negative feedback to obtain a sufficient isolation among various elements. We theoretically present the suitability of the method for small/moderate sized sensor arrays and experimentally verify the predicted behavior by lock-in-amplifier based measurements on a light dependent resistor (LDR) in a 4 × 4 resistor array. Finally, we present a successful implementation of this method on a 16 × 16 imaging array of LDR.     

An Extended Tree-Width Notion for Directed Graphs Related to the Computation of Permanents

It is well known that permanents of matrices of bounded tree-width are efficiently computable. Here, the tree-width of a square matrix M=(m _ij ) with entries from a field \(\mathbb{K}\) is the tree-width of the underlying graph G _M having an edge (i,j) if and only if the entry m _ij ≠0. Though G _M is directed this does not influence the tree-width definition. Thus, it does not reflect the lacking symmetry when m _ij ≠0 but m _ji =0. The latter however might have impact on the computation of the permanent. In this paper we introduce and study an extended notion of tree-width for directed graphs called triangular tree-width. We give examples where the latter parameter is bounded whereas the former is not. As main result we show that permanents of matrices of bounded triangular tree-width are efficiently computable. This result is shown to hold as well for the Hamiltonian Cycle problem.     

Classification of the entangled states 2 × M × N

We extend the matrix decomposition method (MDM) in classifying the 2 × N × N truly entangled states to 2 × M × N system under the condition of stochastic local operations and classical communication. It is found that the MDM is quite practical and convenient in operation for the asymmetrical tripartite states, and an explicit example of the classification of 2 × 6 × 7 quantum system is presented.     

Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier-Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P_NP_M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P_NP_M schemes is that they contain classical high order accurate finite volume schemes (N=0) as well as standard discontinuous Galerkin methods (M=N) just as special cases, which therefore allows for a direct efficiency comparison.In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier-Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P_NP_M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M_s=10.     

The Grid-Pyramid: A Generalized Pyramid Network

The Pyramid network is a desirable network topology used as both software data-structure and hardware architecture. In this paper, we propose a general definition for a class of pyramid networks that are based on grid connections between the nodes in each level. Contrary to the conventional pyramid network in which the nodes in each level form a mesh, the connections between these nodes may also be according to other grid-based topologies such as the torus, hypermesh or WK-recursive. Such pyramid networks form a wide class of interconnection networks that possess rich topological properties. We study a number of important properties of these topologies for general-purpose parallel processing applications. In particular, we prove that such pyramids are Hamiltonian-connected, i.e. for any arbitrary pair of nodes in the network there exists at least one Hamiltonian path between the two given nodes, and pancyclic, i.e. any cycle of length 3, 4 … and N, can be embedded in a given N-node pyramid network. It is also proven that two link-disjoint Hamiltonian cycles exist in the torus-pyramid and hypermesh-pyramid networks.     

Sorting n Numbers on n × n Reconfigurable Meshes with Buses

We show how column sort and rotate sort can be implemented on the different reconfigurable mesh with buses (RMB) architectures that have been proposed in the literature. On all of these proposed RMB architectures, we are able to sort n numbers on an n × n configuration in O(1) time. For the PARBUS RMB architecture, our column sort and rotate sort implementations are simpler than the O(1) sorting algorithms developed in Jang and Prasanna (International Parallel Processing Symposium, 1992) and Lin et al. (Proceedings of Ninth European Workshop on Parallel Computing, 1992). Furthermore, our sorting algorithms use fewer bus broadcasts. For the RMESH RMB architecture, our algorithms are the first to sort n numbers on an n × n configuration in O(1) time. We also observe that rotate sort can be implemented on N × N × ··· × N (k + 1)-dimensional RMB architectures to sort N^k elements in O(1) time.     

Decidable problems on the strong connectivity of Petri net reachability sets

For a Petri net N and a marking M, let R_N(M) be the set of markings reachable from M and let C_N(M) be the set of markings M′ such that M′ ?R_N(M) and M?R_N(M′)C_N(M) is a strongly connected component of R_N(M) to which M belongs. If R_N(M) = C_N (M), then N is said to be M-reversible, and if N is M-reversible for every marking M, then N is said to be reversible. In this paper the following results are presented. (1) C_N(M) is semilinear and therefore it is decidable whether (i) M′ ?C_N(M), (ii) C_N(M) is a finite set, (iii) C_N(M) ? C_N′(M′), and (iv) given two markings M and M′ such that M ? M′, there is a nonnegative integer k such that M + k(M ? M′) ?R_N(M′). (2) It is decidable whether (i) N is M-reversible or not, and (ii) N is reversible or not. (3) Given a Petri net N and a marking M, we can construct an M-reversible Petri net N′ such that C_N(M) = R_N′(M). (4) The equality problem for the sets of all firing sequences of an M-reversible Petri net N and an M′-reversible Petri net N′ is decidable. And some related problems are discussed.     

Short rational functions for toric algebra and applications

We encode the binomials belonging to the toric ideal I_A associated with an integral d×n matrix A using a short sum of rational functions as introduced by Barvinok (Math. Operations Research 19 (1994) 769) and Barvinok and Woods (J. Amer. Math. Soc. 16 (2003) 957). Under the assumption that d and n are fixed, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal I_A, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate applications to enumerative combinatorics, integer programming, and statistics.     

Yang-Baxter {\breve R} matrix, entanglement and Yangian

We present a method to construct ??X?? form unitary Yang-Baxter ${\breve R}$ matrices, which act on the tensor product space ${V_{i}^{j_{1}}\otimes V_{i+1}^{j_{2}}}$ . We can obtain a set of entangled states for (2j ₁?+?1)?× (2j ₂?+?1)-dimensional system with these Yang-Baxter ${\breve R}$ matrices. By means of Yang-Baxter approach, a 8?× 8 Yang-Baxter Hamiltonian is constructed. Yangian symmetry and Yangian generators as shift operators for this Yang-Baxter system are investigated in detail.     

Hamilton-connectivity and cycle-embedding of the Möbius cubes

The recently introduced interconnection network, the Möbius cube, is an important variant of the hypercube. This network has several attractive properties compared with the hypercube. In this paper, we show that the n-dimensional Möbius cube M_n is Hamilton-connected when n?3. Then, by using the Hamilton-connectivity of M_n, we also show that any cycle of length l (4?l?2ⁿ) can be embedded into M_n with dilation 1 (n?2). It is a fact that the n-dimensional hypercube Q_n does not possess these two properties.